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BITSAT 2024Paper 1
Exam
BITSAT
Year
2024
Paper
Paper 1
Subject
Mathematics
Chapter
Complex Number

BITSAT 2024 · Paper 1 · Mathematics · Complex Number

If |z_1|=2,|z_2|=3,|z_3|=4 and |2 z_1+3 z_2+4 z_3|= 4 , then...

If ∣z1∣=2,∣z2∣=3,∣z3∣=4and∣2z1+3z2+4z3∣=\left|z_1\right|=2,\left|z_2\right|=3,\left|z_3\right|=4 and \left|2 z_1+3 z_2+4 z_3\right|= 4 , then absolute value of 8z2z3+27z3z1+64z1z28 z_2 z_3+27 z_3 z_1+64 z_1 z_2 equals

Options

  1. A2424
  2. B4848
  3. C7272
  4. D9696

Correct answer

D. 9696

Step-by-step solution

∣8z2z3+27z3z1+64z1z2∣=∣z1∣∣z2∣∣z3∣∣8z1+27z2+64z3∣=(2)(3)(4)∣8zˉ1∣z1∣2+27zˉ2∣z2∣2+64zˉ3∣z3∣2∣=24∣2zˉ1+3zˉ2+4zˉ3∣=24∣2z1+3z2+4z3‾∣=24∣2z1+3z2+4z3∣=(24)(4)=96\left|8 z_2 z_3+27 z_3 z_1+64 z_1 z_2\right|=\left|z_1\right|\left|z_2\right|\left|z_3\right|\left|\frac{8}{z_1}+\frac{27}{z_2}+\frac{64}{z_3}\right|=(2)(3)(4)\left|\frac{8 \bar{z}_1}{\left|z_1\right|^2}+\frac{27 \bar{z}_2}{\left|z_2\right|^2}+\frac{64 \bar{z}_3}{\left|z_3\right|^2}\right|=24\left|2 \bar{z}_1+3 \bar{z}_2+4 \bar{z}_3\right|=24\left|\overline{2 z_1+3 z_2+4 z_3}\right|=24\left|2 z_1+3 z_2+4 z_3\right|=(24)(4)=96
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